Digital frequency modulated continuous wave radar using handcrafted constant envelope modulation

ABSTRACT

A method for determining a frequency modulation includes generating a symbol stream that is filtered, with multiple samples per period. Sample values represent samples of the filtered symbols at instants separated by intervals of a fraction of a time period between successive symbols. Samples of I and Q waveforms are calculated from frequency modulating a signal with the sequence of symbols. For each possible set of symbol values on which a waveform depends, an average waveform is produced over all symbol values outside the group; and on which the waveform is not to depend, all waveforms are superimposed within +/−half a period of the center symbol of each group having the same set of values and averaging the superimposed I, Q samples to produce for each group an averaged set of samples and an average waveform. Final I, Q values are stored for subsequent frequency modulation.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application is a continuation of U.S. patent applicationSer. No. 15/492,159, filed Apr. 20, 2017, now U.S. Pat. No. 9,945,935,which claims the filing benefits of U.S. provisional applications, Ser.No. 62/469,165, filed Mar. 9, 2017, Ser. No. 62/382,857, filed Sep. 2,2016, and Ser. No. 62/327,003, filed Apr. 25, 2016, which are all herebyincorporated by reference herein in their entireties.

FIELD OF THE INVENTION

The present invention is directed to radar systems, and moreparticularly to radar systems for vehicles.

BACKGROUND

The use of radar to determine range and velocity of objects in anenvironment is important in a number of applications includingautomotive radar and gesture detection. A radar system typicallytransmits a signal and listens for the reflection of the signal fromobjects in the environment. By comparing the transmitted signal with thereceived signal, a radar system can determine the distance to an object.Using multiple transmissions, the velocity of the object can bedetermined. Moreover, using multiple transmitters and receivers, thelocation (angle) of the object can also be determined.

There are several types of waveforms used in different types of radarsystems. One type of waveform or radar signal is known as afrequency-modulated continuous waveform (FMCW). In an FMCW-type radarsystem, the transmitter of the radar system sends a continuous signal inwhich the frequency of the signal varies. This is sometimes called achirp radar system. Mixing (multiplying) a waveform reflected from anobject (also known as a target) with a replica of the transmitted signalresults in a CW signal with a frequency that represents the distancebetween the radar transmitter/receiver and the target. By sweeping up infrequency and then down in frequency, the Doppler frequency can also bedetermined.

There is a continuous need for improved radar techniques that achievegood range performance without excessive transmitter power, which permitmultiple users to share the spectrum, and which achieve an improvedtradeoff between instantaneous bandwidth occupancy and range resolution.

SUMMARY

An FMCW radar system comprises one or more constant envelopetransmitters for transmitting radio signals that are frequencymodulated. The frequency modulation uses codes to deviate the frequencyfrom a mean or center frequency according to one of a limited number ofshaped frequency transitions associated with a limited number ofsuccessive codes. The codes of each transmitter are different andpreferably exhibit low cross-correlation. In one exemplaryimplementation, for each transmitter, the frequency modulated signal maybe produced by expressing the frequency modulation as a sequence ofgenerated I and Q baseband vectors that are dependent on the limitednumber of successive codes and which have a constant envelope propertywhere I²+Q² is a constant, for example, unity. The values are modulatedon to a microwave carrier frequency for transmission by the radartransmitting antenna, for example by using an I,Q modulator. The I and Qwaveforms are precomputed to depend on a limited number (N) ofsuccessive bits of a code, for example, 2 or 3 bits, and the precomputedwaveforms are stored in a memory as numerical values. A plurality of I,Qvalues are stored in memory for each possible pattern of the Nsuccessive bits and a state variable indicative of the phase quadrant.The plurality of values are read from the memory sequentially for eachnew value of a code bit, the memory being addressed by the new bit, N−1previous bits and the state variable. Each plurality of the I,Q valuesis engineered to obtain an optimum compromise between a number of oftenconflicting criteria, including compliance with a spectral mask, rangeresolution, the ease or difficulty of discriminating weak targets fromclose by strong targets, and correlation loss with target echo delays ofa non-integral number of bit periods.

For operating at very high digital code rates, the memory is organizedas a plurality of N memories that are read at the code rate divided byN. Each pair of read I,Q values is digital to analog converted using a Dto A converter that shapes the quantizing noise to reduce its spectraldensity near the microwave carrier frequency, and low-pass filtered toobtain analog I,Q signals that are applied to the I,Q modulator.

In an aspect of the present invention, a radar system for a vehicleincludes a transmitter and a receiver. The transmitter transmits anamplified and frequency modulated radio signal. Each transmittercomprises a frequency generator, a code generator, a modulator, aconstant-envelop power amplifier, and an antenna. The frequencygenerator is operable to or configured to generate the radio signal witha desired mean or center frequency. The code generator is operable to orconfigured to generate a sequence of chips at a selected chiprate. Amodulation interval between successive chips is a reciprocal of thechiprate. The modulator frequency is operable to or configured tomodulate the radio signal such that the frequency modulation comprisesshaped frequency pulses. The shaped frequency pulses correspond to afirst signal, the frequency of which deviates from the desired mean orcenter frequency during each of the modulation intervals according to aselected pulse shape. The constant-envelope power amplifier amplifiesthe frequency modulated radio signal at a desired transmit power level.The antenna transmits the radio signal.

These and other objects, advantages, purposes and features of thepresent invention will become apparent upon review of the followingspecification in conjunction with the drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a plan view of an automobile equipped with one or more radarsystems in accordance with the present invention;

FIG. 2A and FIG. 2B are block diagrams of radar systems in accordancewith the present invention;

FIG. 3 is a block diagram illustrating a flow of data structures througha radar system in accordance with the present invention;

FIG. 4 is a block diagram illustrating a radar system with a pluralityof receivers and a plurality of transmitters (MIMO radar) for producingthe data structures of FIG. 3, in accordance with the present invention;

FIG. 5 illustrates an exemplary transmitter architecture in accordancewith the present invention;

FIG. 6 illustrates an exemplary eye diagram for Gaussian minimum shiftkeying (GMSK) with a BT factor of 0.3 in accordance with the presentinvention;

FIG. 7 is a graph illustrating the correlation of a transmitted signalwith a transmitted code in accordance with the present invention;

FIG. 8 is a graph illustrating an autocorrelation function for GMSK withBT=0.3 in accordance with the present invention;

FIG. 9 is a graph illustrating a power spectrum for GMSK with BT=0.3 inaccordance with the present invention;

FIG. 10 is a graph illustrating correlation sidelobes using a Gaussianreceiver filter with BT=0.3 in accordance with the present invention;

FIG. 11 is a graph illustrating variation of sidelobe levels withreceiver filter BT in accordance with the present invention;

FIG. 12 is a graph illustrating variation of noise bandwidth and SNRwith receiver filter BT in accordance with the present invention;

FIG. 13 is a graph illustrating an eye diagram of modulation engineeredto depend on 3 chips in accordance with the present invention;

FIG. 14 is a graph illustrating the correlation function of themodulation of FIG. 13 in accordance with the present invention;

FIG. 15 is a graph illustrating a comparison of the spectrum of regularGMSK with that of FIG. 14 in accordance with the present invention;

FIG. 16 is a graph illustrating the correlation function of a morehandcrafted waveform in accordance with the present invention;

FIG. 17 is a graph illustrating an eye diagram of the handcraftedwaveform in accordance with the present invention;

FIG. 18 is a graph illustrating the spectrum of the handcrafted waveformin accordance with the present invention;

FIG. 19 is a set of graphs illustrating trellis and constellationdiagrams in accordance with the present invention;

FIG. 20 is a graph illustrating the eye diagram of MSK computed using 16samples per chip in accordance with the present invention;

FIG. 21 is a graph illustrating the spectrum of unfiltered MSK inaccordance with the present invention;

FIG. 22 is a graph illustrating the correlation function of MSK versusthe number samples per chip in accordance with the present invention;

FIG. 23 is a graph illustrating raised cosine digital FM in accordancewith the present invention;

FIG. 24 is a graph illustrating the eye diagram for raised cosinedigital FM in accordance with the present invention;

FIG. 25 is a graph illustrating the spectrum of raised cosine digital FMin accordance with the present invention;

FIG. 26 is a graph illustrating the correlation function for raisedcosine digital FM in accordance with the present invention;

FIG. 27 is a graph illustrating the eradication of correlation sidelobeskirts of MSK with 8 samples/chip by waveform handcrafting in accordancewith the present invention;

FIG. 28 is a graph illustrating the correlation function of handcraftedMSK at 4 samples/chip in accordance with the present invention;

FIG. 29 is a graph illustrating the correlation function for raisedcosine digital FM at 4 samples/bit after handcrafting in accordance withthe present invention;

FIG. 30 is a graph illustrating the variation of parameters withreceiver BT for raised cosine transmitter modulation in accordance withthe present invention;

FIG. 31 is a graph illustrating the parameters of FIG. 30 on a finer dBscale in accordance with the present invention;

FIG. 32 is a graph illustrating a class of shaping functions to beinvestigated in accordance with the present invention;

FIG. 33 is a graph illustrating other polynomial-based frequency-pulseshaping functions in accordance with the present invention;

FIG. 34 is a flow diagram of a processing for simulating modulation inaccordance with the present invention;

FIG. 35 is a graph illustrating one possible eye diagram optimized forfour (4) samples per bit in accordance with the present invention;

FIG. 36 is a graph illustrating the spectrum of the waveform of FIG. 35with Gaussian post D to A filter BT=0.8 in accordance with the presentinvention; and

FIG. 37 is a graph illustrating the correlation sidelobes for thesignals of FIGS. 35 and 36.

DETAILED DESCRIPTION

The present invention will now be described with reference to theaccompanying figures, wherein numbered elements in the following writtendescription correspond to like-numbered elements in the figures. Methodsand systems of the present invention may achieve a good performancerange without excessive transmitter power requires and provide improvedtradeoffs between instantaneous bandwidth occupancy and rangeresolution, through the use of constant envelope transmitter amplifiersand frequency modulation using smoothly shaped frequency deviationpulses.

Small, low-cost radar systems are increasingly becoming of interest formotor vehicle collision avoidance applications. National frequencymanagement authorities such as the FCC in the USA have made availablecertain frequency bands in the millimeter wave region for this purpose,for example the frequency band 76 to 77 GHz and the band 81 to 86 GHz.

Automobile radar systems become of greater utility the greater theobject resolution achieved in ultimately the three dimensions of range,azimuth and elevation, as well as in Doppler shift, which indicatesrelative velocity of a target object. An ultimate goal is objectrecognition and hazard detection using the radar data, possibly infusion with video data, map databases, and GPS positioning.

As with communications systems such as cellular phones, the frequencyband has to be shared by many users without unacceptable mutualinterference, so the same concerns of multiple access efficiency,spectral efficiency and capacity arise, in terms of the number ofdevices per square kilometer that can be simultaneously operated.Through generations 1,2,3 and 4 of mobile phone systems, many differenttechniques of modulation and coding have been explored to optimizecapacity, including Frequency Division Multiple Access (FDMA), TimeDivision Multiple Access (TDMA), Code Division Multiple Access (CDMA)also known as Direct Sequence Spread Spectrum (DSSS) and FrequencyHopping Spread Spectrum (FHSS). Many different modulation methods havealso been explored, including Analog Frequency Modulation (FM), Digitalfrequency modulation, such as GSM's Gaussian Minimum Shift Keying(GMSK), and all the usual digital phase modulation schemes such asQuadrature Phase Shift Keying (QPSK), Offset QPSK (OQPSK), QuadratureAmplitude modulation (QAM, 16QAM, etc.), and latterly OrthogonalFrequency Division Multiplexing (OFDM).

In communications systems operating in the lower microwave frequencies(900 MHz to L-band) and higher (S-band to 2400 MHz), multipathpropagation has increasingly become a problem. For example, transistorfrequency performances have increased to the point where radio devicescan be made economically at much higher frequencies than before.However, signals at shorter wavelengths are reflected by smallerobjects, and such delayed reflections distort digital transmission,causing intersymbol interference (ISI). Higher frequency digitalcellular communication only became possible through the use of advanceddigital signal processing algorithms that could correctly decodeinformation distorted by ISI. Research into such techniques remains thedominant subject of wireless communications and resulted in the mostrecent shift to OFDM.

Unlike communications systems where multiple, differently-delayedreflections are a nuisance, in radar systems, the delayed reflectionsare the wanted information. Also in contrast with communicationssystems, except in bistatic radar systems, the signal reflected from anobject or target and processed by a radar receiver has originated in theradar's own transmitter, which may be in intimate proximity to thereceiver. Thus, the receiver can use information on exactly what wastransmitted, and when, to aid in analyzing the received signal, and todetermine the delays of target echoes which indicate their range.

There are several different types of radar systems. The most well-knownis pulse radar, in which a very short pulse of very high power microwaveenergy is transmitted during which time the receiver is blanked toprevent overload or damage; then the receiver is unblanked and listensfor echoes received with various delays. The length of time the receivercan listen before the next transmitter pulse equates to the maximumrange. The antenna may rotate between pulses to test for reflectingobjects at different azimuths or elevations or both.

A less common variation of the above is the bistatic radar system inwhich the transmitter is not co-located with the receiver and uses atotally different antenna. The receiver thereby does not need to beblanked during the transmit pulse.

In pulse radar systems, the transmitter duty factor and therefore themean power is small; therefore, to achieve enough sensitivity for longrange performance a high peak pulse power must be used. To overcomethat, another type of radar called continuous wave (CW) radar is used. ACW radar transmits and receives all the time. The transmitted signal hasfeatures in its waveform that enable the receiver to determine the delayof a received signal by determining the time difference between thetransmitted feature and the received feature. In FMCW-type radarsystems, the feature used is the instantaneous frequency. Thetransmitter frequency is changed linearly and very rapidly from astarting value to an ending value to create what is known as a chirp. Adelayed signal will be received at an earlier value of the chirpfrequency. By forming a beat between the transmit frequency and thereceived frequency in the receive mixer, and determining the beatfrequency, which is the transmit-receive frequency difference, the delayof the reflected chirp can be calculated. Because such a frequencydifference cannot be distinguished from Doppler, a forward and backwardchirp may be used alternately, producing a sawtooth frequencymodulation. Any Doppler has opposite effect on interpreting the forwardchirp compared to the backward chirp, thus allowing range and Doppler tobe separated. In FMCW radar systems, one issue is the extreme accuracyand linearity needed for the chirp signal. The greatest issue in CWradar is receiving at the same time as transmitting. The transmittedsignal is much stronger than any received echo and can overload thereceiver's limited dynamic range.

Another version of CW radar called pulse-CW radar aims to reduce thedifficulty of receiving weak echoes from distant objects in the presenceof the strong own transmitter signal. This is similar to pulse radarexcept that the transmitter duty factor is much higher, for example 50%.A modulated transmit pulse is transmitted for a duration that fills upthe time to the furthest object and then switches off. Meanwhile, thereceiver attempts to receive strong echoes from nearby objects while thetransmitter is transmitting, but when receiving weak later echoes fromdistant objects, the transmitter has already switched off, facilitatingtheir detection. Improving near-far performance in radar systems isdescribed in detail in U.S. patent application, Ser. No. 15/292,755,filed Oct. 13, 2016 (“the '755 patent”), which is hereby incorporated byreference herein in its entirety.

In the following disclosure, digital codes are sometimes referred to ascomprising a bit sequence and sometimes as comprising a chip sequence.The terms “chips” and “bits” are used interchangeably herein, and meanbinary valued quantities. The binary values are 0 or 1 in Booleannotation or +1 and −1 in numerical notation. They may also beabbreviated to just + and − signs. The term “symbols” is also known, andmay apply to binary values or multi-valued quantities selected from afinite alphabet. When a multi-valued quantity can exhibit 2^(N)different values, it can also be equated with N binary values or bits.Therefore, it should be understood that grouping a number of bits into amulti-valued symbol and describing a system in terms of symbols ratherthan bits or chips does not represent a significant technical departurefrom the teachings herein. Also, while the invention is described interms of waveforms that have four principal constellation points of+/−90 degrees and 0/180 degrees, a person of normal skill in the artwould be able to produce variations using the teachings herein that usedhigher order constellations such as 8-PSK or MFSK. Hereinafter, theinvention shall be described in terms of binary bits or chips, but thescope of the invention encompasses all such variations as may be made bya person of normal skill in the art and described by the attachedclaims.

The invention is described primarily for use in a digital FMCW radar inwhich transmission and reception occur simultaneously at a same site.However, the modulation is also useful in digital FM radars that do notnecessarily transmit and receive at the same time, but ratheralternately. Hybrid radars can also be made in which transmission andreception are simultaneous for a first period and then the transmitterswitches off to allow the receiver to receive weak, late echoes withoutstrong interference from the local transmitter, as discussed in the '755patent.

All such references to digital FMCW radar therefore, especially in theclaims, shall also be interpreted to encompass the above variations,unless explicitly limited by appropriate wording.

In all radar systems, the distance resolution is ultimately related tothe width of the autocorrelation function of the transmitted signal.Advanced algorithms such as Multiple Signal Classification (MUSIC) allowresolution less than, but still related to the width of theautocorrelation function.

The power spectrum is the Fourier transform of the autocorrelationfunction and so has a spectral occupancy inversely proportional to therange resolution. When a signal with certain spectrum S(jw) istransmitted and received with a matched filter H(jw) that has theconjugate frequency response to that of the transmit spectral shaping,namely H(j(w)=S(−jw), the output has a spectrum that is shaped by theproduct of the transmit shaping and its conjugate at the receiver,namely by S(jw)S(−jw)=|S(jw)|², which is the power spectrum shape, andthus has a correlation function equal to the autocorrelation function ofthe transmitted signal. However, in a practical realization, thereceiver does not necessarily receive the transmitted signal with amatched filter, so deviations in the relationship between rangeresolution and signal autocorrelation function may arise. In that case,the correlation curve exhibited when the receiver correlates a receivedsignal with a transmitted chip sequence that is received with variousdelays must be computed versus the delay for each case, and is hereintermed the correlation function. With small delays of plus or minus twoor three chips, the shape of the correlation function mimics the impulseresponse of the entire channel that exists between the transmitter'scode generator and the point at which the received signal is extractedinto the correlator. For large relative shifts of many chips or bits,the correlation function will exhibit the autocorrelation function ofthe digital code chosen. It is well known that Maximum Length Sequencesexhibit autocorrelation functions that only have one large peak, and allsidelobes are at a level relative to the peak of −1/N, where N is thelength of the code. If they can be used, this autocorrelation propertyis a desirable one for radar systems.

As noted above, FMCW radar typically used chirp signals to determinerange and Doppler.

A digital FMCW radar on the other hand transmits an RF signal which isfrequency modulated with a digital code sequence to produce atransmitted signal that has good autocorrelation properties thatfacilitate range resolution while exhibiting good spectral containment.One type of frequency modulation that appears to have interestingproperties in this regard is minimum shift keying (MSK). In MSK, thefrequency is changed between two values spaced at plus and minus onequarter (¼) the chiprate (=bitrate) from the carrier, with the resultthat the phase, which is the integral of frequency deviation, changes by+/−quarter (¼) of a cycle over each bit or chip. Thus, at the end ofeach chip, the signal vector lies at one of two diametrically oppositepoints along a line at right angles to the prior signal vector position.MSK is related to Offset QPSK in that the signal vector for even bitsends up at +/−1 while the signal vector for odd bits ends up at +/−j.The difference however is, that when the 1<-->0 transitions of thedigital code are filtered or shaped to contain the spectrum, the MSKsignal remains at a constant amplitude while an OQPSK signal acquiresamplitude modulation, requiring a linear transmit power amplifier topreserve it. Such linear power amplifiers have lower efficiency thanconstant envelope amplifiers because they do not operate at the optimumpower point 100% of the time. At low microwave frequencies such asL-band and S-band, a solid state constant envelope transmitter mayachieve 60% efficiency while a linear power amplifier may achieve only30% efficiency. Since even class-C constant envelope solid-statetransmit power amplifiers operating at millimeter wave frequencies onlyhave efficiencies of the order of 15% at the present state of the art,the extra loss of efficiency of a linear power amplifier is to beavoided. Thus, constant amplitude phase modulations such as MSK are ofgreat interest for digital FMCW radar use.

The bandwidth of the transmitted RF signal using digital FM isproportional to the chiprate of the digital modulating code, while therate at which the spectrum falls off outside of the main spectral lobedepends on the shaping applied to the frequency modulation. It is wellknown that filtering an MSK modulating waveform using a Gaussian filterproduces, for some coincidental reason, the greatest ultimate rate ofspectral fall-off outside the main occupied bandwidth. This wasparticularly exploited for the GSM digital cellular phone system whichemployed this modulation, termed Gaussian minimum shift keying (GMSK).In this application, versions of such modulations are described that areparticularly optimized to meet criteria important in the radarapplication, rather than criteria important in the communicationsapplication, and other advantageous frequency modulation pulse shapesare disclosed.

Radars with a single transmitter and a single receiver can determinedistance to a target but cannot accurately determine a direction or anangle of a target from the radar sensor or system unless the antennapattern is steered between pulses either mechanically or electronicallyusing a phased-array. To acquire angular information for each radarpulse period, which in the case of the exemplary radar system describedherein comprises a sequence of frequency modulating bits with which thereceiver performs correlation, either multiple transmitter antennas ormultiple receiver antennas or both are needed, and which are operativein all directions all the time. Each receiver receives and separateseach echoed transmitter signal, thus resulting in N×M received results,where N is the number of transmitters and M is the number of receivers.With proper design, these N×M results can be post-combined in any numberof ways according to a plurality of beamforming vectors, therebyachieving elevation and azimuth location of each signal as well as rangeand Doppler information.

The larger the number of transmitter antennas and receiver antennas, thebetter the resolution possible. Each transmission antenna is connectedto a separate transmitter, and each receiver antenna is connected to aseparate receiver. As discussed herein, such a radar system is known asa multiple-input, multiple-output (MIMO) radar system.

An exemplary MIMO radar system is illustrated in FIG. 4. With MIMO radarsystems, each transmitter signal is rendered distinguishable from everyother transmitter by using appropriate differences in the modulation,for example, different digital code sequences. Each receiver correlateswith each transmitter signal, producing a number of correlated outputsequal to the product of the number of receivers with the number oftransmitters. The outputs are deemed to have been produced by a numberof virtual receivers, which can exceed the number of physical receivers.A receiver may be referred to as a virtual receiver even when there isonly a single transmitter, in order to avoid changing the terminology.The output of a given receiver receiving a transmitted signal from agiven transmitter has a phase that depends on the loop distance from thetransmitting antenna to the receiving antenna. Each transmit-receivecombination produces a different loop phase due to the separation oftheir antennas. By combining the outputs for each transmitter/receivercombination while correcting for these different loop phase shifts, acombined output is obtained that only constructively adds for a targetat a unique point in space. By repeating the combination using theprecomputed loop phase shifts for many different points in space,signals may be resolved in the three dimensions of range, azimuth andelevation. The focusing effect of the above phase coherent combining iseffective for resolution in azimuth and elevation but only contributesto range resolution at very short ranges, and the range resolution atlong ranges is principally determined by the round-trip delay of thedigital modulation. An exemplary radar system according to the inventiontherefore determines the range of a target or the distance to a targetprincipally by determining how long it takes an echo of transmitted RFsignals to be heard back at the receivers. From this measured time-delayand knowing that the electromagnetic RF signals travel at the speed oflight (or ultrasonic signals traveling at the speed of sound), thedistance can be determined.

In digital FMCW radar, the method of determining the time delay is bycorrelating a received RF signal with multiple time-shifts of thedigital modulating code to produce correlations which are stored inrange bins. The length of time over which coherent correlations can beperformed is limited by the phase rotation caused by Doppler shift. Tocontinue cumulative correlation for longer times than this, partialcorrelations are combined while compensating for the Doppler-inducedphase drift. The partial correlations may be stored for each virtualreceiver and range in a 3-dimensional array called a radar data cube, asillustrated in FIG. 3, in which the three dimensions are virtualreceiver number, range, and time or index of the partial correlation.Partial correlations for the same receiver and range are then submittedto an FFT, which combines them in a computationally efficient mannerwith many different hypotheses of the rate-of-change of phase, thusproducing long correlations for each of a number of Doppler bins. Theresult is then stored in a radar data cube having the dimensions ofvirtual receiver number, range and Doppler shift. Thus, the radar datacube time dimension has been converted into a Doppler shift dimensionwhich is more meaningful for characterizing a reflecting target orobject as stationary or moving. Then, for the same range and Dopplerbin, the results across different virtual receivers may be combined byusing beamforming matrices as mentioned above in order to achieveangular resolution in azimuth, elevation or both.

Because there can be multiple objects in the environment, there will bemultiple bins in the radar cube for which there will be a highcorrelation. While a virtual receiver/radar could correlate the receivedRF signal with all possible delays, generally there is a finite set ofdelays with which the virtual receiver/radar will correlate, that is, afinite set of range bins over the range of interest. Likewise, therewill be a finite set of Doppler bins up to the maximum conceivablerelative velocity between the radar and an oncoming vehicle. Because thetransmission and return range changes at twice the relative velocity ofthe target to the radar, the maximum Doppler shift may be based on fourtimes the maximum speed of any one vehicle. For a maximum vehicle speedof 250 km/hr, which can be reached on the German Autobahn for example,the maximum Doppler shift can be that of a 1000 km/hr object, which is74 KHz at 80 GHz. If a radar system's own velocity, which is presumed tobe known, is digitally removed by applying a systematic phasede-twisting to the received data, the maximum Doppler shift drops to 37KHz.

The radar sensing system of the present invention may utilize aspects ofthe radar systems described in U.S. Pat. Nos. 9,772,397; 9,753,121;9,575,160; and/or 9,599,702; and/or U.S. provisional applications, Ser.No. 62/382,857, filed Sep. 2, 2016, Ser. No. 62/381,808, filed Aug. 31,2016, Ser. No. 62/327,003, filed Apr. 25, 2016, Ser. No. 62/327,004,filed Apr. 25, 2016, Ser. No. 62/327,005, filed Apr. 25, 2016, Ser. No.62/327,006, filed Apr. 25, 2016, Ser. No. 62/327,015, filed Apr. 25,2016, Ser. No. 62/327,016, filed Apr. 25, 2016, Ser. No. 62/327,017,filed Apr. 25, 2016, Ser. No. 62/327,018, filed Apr. 25, 2016, and/orSer. No. 62/319,613, filed Apr. 7, 2016, which are all herebyincorporated by reference herein in their entireties.

FIG. 1 illustrates an exemplary radar system 100 configured for use in avehicle 150. A vehicle 150 may be an automobile, truck, or bus, etc. Asillustrated in FIG. 1, the radar system 100 may comprise one or moretransmitters and one or more receivers 104 a-104 d which can be usedjointly to realize a plurality of virtual radars. Other configurationsare also possible. FIG. 1 illustrates a radar system 100 comprising oneor more receivers/transmitters 104 a-104 d, control and processingmodule 102 and indicator 106. The receivers/transmitters 104 a-104 d areplaced to acquire and provide data for object detection and adaptivecruise control. The radar system 100 (providing such object detectionand adaptive cruise control or the like) may be part of an AdvancedDriver Assistance System (ADAS) for the automobile 150.

FIG. 2A illustrates an exemplary radar system 200 with an antenna 202that is time-shared between a transmitter 206 and a receiver 208 via aduplexer 204. As also illustrated in FIG. 2A, output from the receiver208 is received by a control and processing module 210 that processesthe output from the receiver 208 to produce display data for the display212. The control and processing module 210 is also operable to produce aradar data output that is provided to other control units. The controland processing module 210 is also operable to control the transmitter206.

FIG. 2B illustrates an alternative exemplary radar system 250 with apair of antennas 202 a, 202 b: an antenna 202 a for the transmitter 206and another antenna 202 b for the receiver 208. While pulse radarsystems may use shared or separate antennas, continuous-wave radars(discussed herein) will use separate antennas (for transmitting andreceiving) because of their continuous operation. Despite usingdifferent antennas, local spillover from transmitter to receiver is ahuge signal having a short delay. A critical issue in CW radar is theremoval by subtraction of this large local spillover signal, for thesuccess of which an accurately defined modulation, as is disclosedherein, is essential.

FIG. 4 illustrates an exemplary digitally-modulated continuous-waveradar system 400. Radar system 400 comprises a plurality of receiversand their respective antennas 406 and a plurality of transmitters andtheir respective antennas 408. The radar system 400 also includes aflash memory 412, and optionally a random access memory 410. The randomaccess memory 410, for example, an external DRAM, may be used to storeradar data cube(s) instead of using the limited internal (on-chip)memory (e.g., SRAM), and may also be used to store selected range binsfrom a greater number of radar data cubes for post processing to improveDoppler resolution or range resolution by tracking objects using Kalmanfiltering. The radar system may also include a variety ofinterconnections to an automotive network, e.g., Ethernet, CAN-FD,and/or Flexray.

As was indicated above, range resolution is related to the width of theautocorrelation function of the transmitted signal. A practicalautocorrelation function width cannot be too small, otherwise it willhave to be computed from the received signal with a sufficiently highsampling density to avoid missing the peak, and the results have to bestored in memory for further analysis, e.g. Doppler analysis. Therefore,computational power and on chip memory limitations, or, in the case ofoff-chip memory, I/O bandwidth limitations, limit the narrowness of theautocorrelation function that can be contemplated. In a digital FMCWradar system based on transmitting digital codes, one possible samplingdensity is one sample per chip period, obtained by correlating thetransmitted sequence with different whole-chip shifts of the receivedsignal. It could be contemplated to correlate with half-chip shifts ofthe received signal, but if sufficient memory is available to store thatdouble number of results, then the chiprate may as well be doubled toreduce the width of the autocorrelation function, if bandwidth isavailable. In the present application, bandwidth in the 80 GHz range isnot the limitation. Therefore, the practical solution is to determinehow many correlations per second can be computed and stored, and toequate that with the chip rate, such that correlations are to becomputed only for whole-shifts of the received signal. Therefore, thecharacteristics of the autocorrelation functions, computed at whole-chipshifts, need to be investigated for digital code frequency-modulatedsignals. Several known algorithms exist for computing many correlationsbetween one or more codes and multiple shifts of a received signal; forexample, a technique using FFTs for performing circular convolution isknown.

FIG. 5 illustrates an exemplary transmitter block diagram. A digitalcode generator 1010 is fed with a chiprate clock to produce apseudorandom code for modulating the transmitter. The pseudorandom codepreferably has good autocorrelation sidelobe properties at least up to ashift corresponding to the round-trip delay to a target at maximumrange. The pseudorandom codes used by different transmitters of a MIMOradar should also be preferably mutually orthogonal or have lowcross-correlation. In a first implementation of a radar embodying theinvention, the codes are merely random and a long correlation lengthrelied upon to reduce cross correlation and autocorrelation to the pointwhere subtractive interference cancellation can take over and furthersuppress strong targets to reveal weaker targets. Optionally, thesequences of random binary values (codes or chips) may be provided by atruly random number generator or a pseudorandom number generator. Suchnumber generators are explained in more detail in U.S. patentapplication, Ser. No. 15/204,003, filed Jul. 7, 2016, which is herebyincorporated by reference herein in its entirety.

The digital chip code from generator 1010 is fed to I,Q waveformselection logic module 1020, the purpose of which is to select the I,Qwaveform to be modulated for the current chip period in dependence onthe current chip and the chip history, in order to produce a signalhaving a signal frequency varying according to a predetermined shapingfunction. In one implementation, the number of possible waveforms islimited to eight, and thus requires three address bits (a0, a1, and a2)from selection logic 1020 to address waveform memory 1030. In anotherimplementation, the waveform depends on fewer than three successivechips. Each waveform may be described by a number of complex I,Q samplevalues, such as 4, 8, or 16 samples per chip. Counter 1040 is driven bya sample rate clock which is correspondingly 4, 8, or 16 times the chiprate clock that is used to select each I,Q sample pair in turn from thememory. Counter 1040 may be a “divide by 4” using two flip-flops for thecase of 4 samples per chip, a 3-stage divider for 8 samples per chip, ora 4-stage divider for 16 samples/chip. The divided, down-sample rateclock is the desired chip rate clock and may be used to clock thedigital code generator 1010. Each stage of counter 1040 produces adigital output as a further address bit to memory 1030. In the case of 8samples/chip, three counter bits (t0, t1, and t2) are provided to memory1030 to select one of the 8 samples. The selected sample (0 to 7) ofwaveform (0 to 7) comprises a digital I and Q value with a word lengthin the range of 8 to 16 bits. The digital I and Q values are fed intorespective I and Q digital to analog converters (DAC) (1050A, 1050B)where they are converted to analog voltages or currents. At very highspeeds, it is desirable that high speed analog signals be balanced, asthe quality of an on-chip ground cannot be relied upon for single-endedsignals at very high frequencies. The balanced analog I and Q voltagesfrom respective DACs (1050A, 1050B) are then smoothed using respectivelow-pass filters (1060A, 1060B) which may be deliberately engineered, ormay be a collection of incidental bandwidth restrictions produced due tocomponent frequency response limitations. Either way, the filteringneeds to be sufficient to contain the transmitted spectrum to meet theout-of-band limits specified by the frequency management authority. Thefiltered balanced I,Q signals then modulate quadrature carrier signalsproduced by quadrature local oscillator (QLO) 1070 (may also be referredto as a frequency generator) using a pair of balanced modulators (1080A,1080B) (may also be referred as I,Q modulators). The quadrature localoscillator may also be referred to as a carrier frequency generatoroperable to generate a carrier signal that is frequency modulated by thepair of balanced modulators (1080A, 1080B). Furthermore, the I, Qwaveform selection logic 1020, the waveform memory 1030, thedigital-to-analog converters 1050A, 1050B, the low pass filters 1060A,1060B, and the balanced modulators 1080A, 1080B may be collectivelyreferred to as the modulator. Gilbert-cell mixers using 28 nm MOSFETtransistors have proven capable of modulating an 80 GHz carrier signalwith 2 GB digital code rates. Gilbert cell mixers driven by similar QLOsmay be used in the radar receiver in order to produce zero-IF, homodynereceivers.

In a MIMO system, all transmitters and receivers preferably have a knownphase relationship in order to allow the receiver outputs to becoherently combined by beamforming matrices. In one implementation, thedesired phase relationship is guaranteed by injection-locking eachtransmitter and receiver's QLO (frequency generator) to a commonstandard. For a millimeter wave radar operating around 80 GHz, thecommon standard may be a sub-harmonic of the desired millimeter wavefrequency, such as ⅕th or 16 GHz, at which frequency it is easier tofabricate an accurate digital frequency synthesizer or generator to giveprogrammable center or mean frequencies.

The modulated signal at the radar carrier frequency is amplified to atransmit power level in constant envelope power amplifier (PA) 1090which also operates push-pull (i.e. balanced). The push-pull PA 1090 iscross neutralized to reduce Miller feedback, improve the high frequencygain and reduce the S12 parameter. On-chip Balun transformer 1095 may beused to convert the push-pull signal to single ended to bring the signaloff-chip through a ball-bond surrounded by grounded ball-bonds.

FIG. 6 is a graph illustrating the typical eye diagram of the I and Qparts of the transmitted carrier using Gaussian minimum shift keying(GMSK) when no limit is placed on the number of stored waveforms; thatis, each transition can depend on as much past history as the impulseresponse length of the selected Gaussian filter exhibits. For FIG. 6, aGaussian filter with BT=0.3 was used. A filter with BT=1 means the −3 dBbandwidth is equal to the chiprate, and BT=0.3 means the 3 dB bandwidthis 0.3 of the chiprate. Moreover, in common with the definition of GMSKused in the GSM system, the Gaussian premodulation filter is not fedwith square waves (with a polarity equal to the bit values), but byimpulses of unit area (with polarities given by the bit values). Asquare wave bit stream already has an implied sin(x)/xfrequency-response shaped filter that shapes the waveform from impulsesto square waves, and it is desired to investigate the performance offilters that are not constrained to include this inherent sin(x)/xfactor.

FIG. 34 illustrates a flow chart that was used to compute the waveformsof FIG. 6 and subsequent graphs. An 8-bit linear feedback shift registerwas used to generate a maximum length sequence of 255 bits at step 1,and the sequence extended to 256 bits by adding one more bit to obtain apower of 2 bits. If the added bit is a zero, the extended 256 bitsequence will have an equal number of 1's and 0's.

At step 2, each bit is placed in the center of a group of NSPB sampleswith the other samples zero, to represent an impulse having the desiredbit polarity. The bit values are multiplied by NSPB to give the impulseunit area. The number of samples per bit is also chosen to be a power of2, that is 4,8,16,32,64,128 or 256, so that the total number of samplesis a power of 2 equal to 1024,2048,4096,8192,16384,32768 or 65536. Thepurpose is to allow the use of a fast, base-2 FFT at step 3 to producethe spectrum of the unfiltered impulse waveform. At step 4, the spectrumis weighted by the frequency response of the shaping filter, e.g. aGaussian filter. The filtered frequency modulating waveform must beintegrated to obtain the phase waveform. This is conveniently done instep 5 while the signal is still in the frequency domain by dividingeach spectral line by j times its own frequency. When the whole sequenceis 256 bits long, the line spacing is 1/256 of the bitrate, so thefrequency of each spectral line is simply determined. At step 6, aninverse FFT produces the time waveform from the filtered and integratedspectrum.

At any point before step 7, a suitable scaling is applied so that thedesired frequency deviation or modulation index is obtained. At step 7,the I,Q waveform is computed by taking the cosine and sine of the phasemodulation samples, which also has the effect of reducing the phasesmodulo-2π. At this point, if the I,Q waveform does not join upend-to-end, a phase slope may be applied across the I,Q waveform toforce it to join up end to end. This is equivalent to a very smallfrequency shift which can be used later if necessary to ensure that anyfiltering is correctly centered. The purpose of ensuring end-to-endcontinuity is that the FFT at step 10 assumes a cyclic waveform, withoutwhich artifacts may appear on the spectral sidelobes so calculated.

At step 8, the eye diagram is plotted, and manually a phase adjustmentis determined that brings the eye diagram into focus such that maximumeye-openings of the I and Q bits are obtained and so that alltrajectories converge to the minimum number of different I and Qwaveforms.

At step 9 the correlation function and autocorrelation functions may becalculated and displayed, and then the spectrum calculated at step 10 isdisplayed at step 11.

The flow chart may be extended to add other filtering such as the lowpass filters 1060A and 10608 of FIG. 5, and receive filtering, and theeffect on eye diagram, correlation function, and spectrum may becalculated and displayed as at earlier steps.

Note that in order to display an eye diagram having the best eyeopenings that best indicate the values of the modulating symbols, it maybe necessary to determine a common phase rotation to be applied to allI,Q values to remove any phase shift that may be an artifact of theprogram simulating the modulation. In the simulation program used toproduce the results herein, this common phase artifact was determined tobe equal to ⅝ths radians, and so the waveform was rotated by −⅝thsradians at step 8 to display the best eye openings.

After applying the above phase shift, by inspection there appear to beeight possible I,Q waveforms except at the position marked X on therising flank and its corresponding position on the falling flank. Thenumber of waveforms is simply determined by drawing a vertical line atany time point and counting the number of distinct trajectories thatcross it. The number is four each for I and Q near the center of theI-eye or Q-eye for a total of eight, but they diverge slightly to 16 inthe vicinity of point X. That means that the waveform at point X dependson four bits, while the waveform at other times depends only on threebits. The best phase shift mentioned above will be found to reduce thenumber of trajectories to a minimum by converging trajectories that wereotherwise apparently divergent due to the phase shift produced by themodulating program. Manually adjusting the phase shift while observingthe eye diagram will be seen to bring the picture into focus.

The number of bits on which the waveform depends can also be seen bycomputing a correlation between the output waveform and the modulatingchip code. At this point it is necessary to explain the relationshipbetween the digital code chip polarities and the polarity of the I valueat the maximum I-eye opening and likewise the polarity of the Q-value inthe center of its eye. When digital frequency modulation using MSK orGMSK is employed, each modulating bit polarity determines whether theI,Q vector rotates clockwise or anticlockwise by 90 degrees over the bitperiod. Thus, after two bit periods, the I,Q vector will have rotated byeither 0 or 180 degrees. In between, the vector lies at either 90 or 270degrees. Thus, the polarity of even bits determines whether the vectorwill end up at 0 or 180 degrees and the polarity of odd bits determineswhether the vector will end up at 90 or 270 degrees. However, the effectis cumulative, as shown in the table below:

1 0 0 1 1 1 0 0 90 0 −90 0 90 180 90 0

In the above table, bit number 1 is a 1, sending the phase clockwisefrom an assumed zero starting value to +90 degrees. The second bit is azero, sending the phase counterclockwise 90 degrees back to 0. Bit 3 isa 0, sending the phase 90 degrees counterclockwise to −90, and so forth.Thus, the relationship between phase and frequency-modulating bitsequence is:

ϕn=mod2π[Σ πBi/2], i=0 to n   (1)

This may also be written as:

ϕn=mod2π[ϕn−1+πBn/2]  (2)

Thus, if a 2-bit state variable is used to keep track of where the phaseended up last time (that is, the value of ϕn−1), then the phase at theend of the current period can be determined from equation (2).

In the GSM digital cellphone communications system, a simplerrelationship between I,Q polarities and modulating bits was arranged bythe use of precoding. If the desired modulating chip code is designatedCi, then modulating bits Bi are derived from the desired modulating chipcode Ci, according to the precoding equation (3) below:

Bi=C _(i).xor.C _(i−1)   (3)

The relationship between Bi, Ci, phase, and I,Q peak value polaritiesmay be seen in the following table:

C 1 0 1 1  0 1 1 1 0 B 1 1 1 0  1 1 0 0 1 Phase 90  180 270  180 270  0−90  −180 −90  I + jQ j −1 −j  −1 −j 1 −j  −1 −j  −j^(n)(I + jQ) 1 −1 11 −1 1 1 1 −1 

The final row is derived by multiplying the penultimate row of I+jQvalues by −(j)^(n). The result of this systematic progressive phasetwist, which increases at 90 degrees per bit, is to throw the Q valuesup into the real plane, making all values real and in agreement with theoriginal C-values. In GSM, this progressive twist is applied at thereceiver so as to reproduce the code C generated at the transmitter.Without the progressive twist, it may be seen from the I+jQ values thatthe sign progression of I bits is

−1 −1 1 −1 compared to the corresponding bits of C −1 1 1 1

showing that there is a sign alternation. The same is true for Q bits

j,−j,−j,−j,−j compared to 1,1,−1,1 −1

Therefore, an alternative method of ensuring that the signs of Ialternating with Q at the receiver correctly reproduce the intended codeC would be to flip the signs of the C bits at the transmitter accordingto the pattern:

+ + - - + + - - + + - - + + - -

Consequently, there are optional methods for ensuring that the chipsequence C produced by the code generator 1010 at the transmitter isreproduced at a point in the receiver chain where it can be correlatedwith a locally generated replica of C. If this is not done, thencorrelation at the receiver must use the expected signs of I and Q. Theautocorrelation sidelobe characteristics when using the latter methodwill not be the same as the autocorrelation characteristics of code C,but of code C with bits flipped according to the above alternating signpattern. To obtain autocorrelation characteristics intended by design,it is necessary to ensure that the receiver correlates with a codehaving the desired characteristics, and this is ensured by the use ofappropriate transmitter precoding in the I,Q waveform selection logicunit 1020 paired with the correct signal treatment at the receiver. Themethod chosen for the transmitter, i.e. any desired precoding, is builtinto I,Q waveform selection logic 1020.

Using GMSK, the frequency pulse shaping produced by the Gaussian filtermay have an effect beyond the current chip. The phase change producedover one chip period by GMSK frequency pulses may fail to reach 90degrees over one chip period, but when integrated over all chip periodsaffected by a given chip, the cumulative phase change to the signalproduced is exactly +90 degrees so that the four principal terminalpositions of the signal vector remain fixed and do not slowly rotate.This characteristic is maintained in this invention for all frequencypulse shapes considered by constraining the area integral of a frequencypulse shape over its entire impulse response length to be a fixed value.

When the receiver is on the same silicon chip as the transmitter, thelocal replica for correlation is simply derived from the code generator1010 by delaying it if necessary.

When the precoding of equation (3) is applied at the transmitter, andthe receiver applies a systematic progressive 90 degree per chip twistto received signal samples, the twisted samples may be correlated withthe shifts of the code C produced by the code generator 1010. If thereceiver samples the received signal at N samples per chip, thenselecting samples (e.g., 0, N, 2N, 3N), progressively twisting thesamples and correlating with shifts of the code C, produces points onthe correlation function (e.g., 0, N, 2N, 3N). Then, selecting points 1,N+1, 2N+1, 3N+1 etc., progressively twisting them and correlating withC, produces points 1, N+1, 2N+1, 3N+1 etc. of the correlation function.Continuing in this way produces the correlation function for allrelative time shifts in steps of 1/N of the chip period.

The above described correlation function is illustrated in FIG. 7. Themagnitude is plotted in dBs, and the horizontal divisions are one chipperiods. This function is the correlation of the transmitted signal withthe corresponding digital code, or equivalently it is the correlation ofthe received signal with the transmitted code when the receiver does notadd additional filtering. Such a wideband receiver is undesirablehowever, as a bandpass filter is required to limit noise.

The correlation function of FIG. 7 illustrates that, for sampling in thecenter of the eye, the peak correlation is unity as it just reproducesthe mean power of the signal which has been set to unity, and thecorrelation at +/−1 chip period is about −14 dB. The correlation at +/−2chip periods is −80 dB, so the transmitted signal depends substantiallyonly on 3 consecutive bits when sampled at the center of the eye. Thisis exactly in correspondence with what can be seen in in FIG. 6. Ifhowever, the signal is mis-sampled by half a chip period, thecorrelation magnitudes are shown in the table below:

−1.5 chips −0.5 chips +0.5 chips +1.5 chips −39 dB −4 dB −4 dB −39 dB

The above table shows that a 0.5 (one half) chip mis-sampling results ina signal that depends on 4 chips. The correlation +/−2.5 chips away ishowever zero. This also corresponds with what may be seen in FIG. 7 atpoint X, where the signal has 16 different trajectories corresponding toa dependence on 4 chips.

The receiver however cannot remain wideband. The noise bandwidth must belimited. One way of limiting the bandwidth is to use a matched filter,which is known to achieve maximum signal-to-noise ratio. A matchedfilter corresponds to correlating the signal with the complex conjugateof itself. This produces the autocorrelation function (ACF). The ACF forthe same signal is shown in FIG. 8.

FIG. 8 shows that with optimum sampling, the ACF at +/−1 chip period isabout −7 dB and is +/−31 dB at +/−2 chip periods. The signal nowtherefore depends somewhat on five consecutive chips due to theadditional ISI introduced by matched filtering. With half a chipmis-sampling, the ACF has the values shown in the table below.

−2.5 chips −1.5 chips −0.5 chips +0.5 chips +1.5 chips +2.5 chips −53 dB−17 dB −2 dB −2 dB −17 dB −53 dB

With half a chip mis-sampling, the signal is showing a dependence oneither 4 chips or 6 chips, depending on whether a correlation level of−53 dB is significant in the application.

In a communications system, correlation values on the order of −30 dBare not of significance because they do not significantly affectinformation error rates. In a radar system however, a strong target echocan easily be 30 dB above a weak target echo two chips away. Therefore,achieving low autocorrelation sidelobes is of greater importance inradar applications. If autocorrelation sidelobes remain high severalchips away, strong target subtraction may then be necessary to revealweaker target echoes with neighboring ranges. The complexity of strongtarget subtraction may therefore be reduced or eliminated entirely ifcorrelation sidelobes can be adequately suppressed.

A number of ways of reducing autocorrelation sidelobes will now bediscussed. Firstly, it may be acceptable to use a slightly wider filterthan the matched filter in the receiver. To get an idea of suitablereceiver bandpass filter bandwidths, the spectrum of the signal shown inFIG. 9 is used. The horizontal divisions are 0.5 times the chiprate. The−3 dB bandwidth is approximately +/−0.25 times the chiprate, about thesame as the Gaussian premodulation filter. Therefore, a Gaussian filterof the same BT factor can be contemplated for the receiver filter.

FIG. 10 illustrates correlation sidelobes using a Gaussian receivefilter with BT=0.3. There is a 1.05 dB loss of power though such afilter and a 2 dB loss in peak correlation. However, the noise bandwidthof the filter is only 0.63 bitrates, which is a reduction of 1.95 dB,substantially compensating for any loss of signal power and correlationmagnitude. The signal-to-noise ratio is therefore about the same as witha matched filter correlator. The sidelobes however are now reduced from−31 dB at +/−2 chips, using the matched filter, to −37 dB relative tothe peak of correlation. The signal-to-noise ratio effects and thecorrelation sidelobes can now be explored as a function of thereceiver's BT factor. FIG. 11 illustrates how the sidelobe levels at+/−0.5 chip, +/−1 chip, +/−1.5 chips, +/−2 chips and +/−2.5 chips dependon receiver filter BT. Also, the noise bandwidth and peak correlationloss are shown, and combined to show the SNR loss involved in choosinghigher receiver BT factors to reduce correlation sidelobes. Thecorrelation at +/−0.5 chip has the practical significance that itrepresents the loss of peak correlation that occurs due to a target echoarriving with a delay that is a non-integral number of chip periods.FIG. 12 shows the latter as well as noise bandwidth, peak correlationloss, and net SNR loss on a finer dB scale.

Many other filter responses could also be explored, such as boxcarfilters, Bessel filters and the like, and the number of cases that canbe explored are too numerous to address in this application, the purposeof which is directed more towards choice of modulation, which is atransmitter question rather than a receiver question. Attention istherefore turned to what can be done on the transmitter side to reducecorrelation sidelobes.

FIG. 7 shows that the correlation function for GMSK with BT factor=0.3essentially has a 5-chip spread. A 4-chip spread is evident from the 16trajectories visible in FIG. 6, but the 5th chip dependence is there,only too small to be visible by the naked eye on the eye diagram. Wereit not for the divergence of the trajectories at location X in FIG. 6,there would be no more than 8 trajectories and therefore only 3 chips ofcorrelation spread. The waveform may therefore be handcrafted toconverge these trajectories to a total of 8.

The first step in handcrafting the waveform is to compute, for a givengroup of three chips within +/−1 chip of a waveform point, the averageof all waveform values over the four other combinations of the two chipsat +/−2 chips away. The average waveform points are intended to bestored at a given number of samples per chip in waveform memory (1030)of FIG. 5 for all 8 combinations of the group of three chips, andselected from memory when those 3 chip values are presented as addressa2,a1,a0 from waveform selection logic (1020). Now the waveforms exhibitmany symmetries, such as I/O symmetry, time-reversal symmetry, and+/−symmetry, but for very high chiprates such as 2 GB/s it can be moreburdensome to try to exploit those symmetries to reduce the memory sizethan to merely accept the full memory size. At lower chiprates,exploiting the symmetries might result in a net reduction of siliconarea.

FIG. 12 illustrates the eye diagram of GMSK with BT=0.3 when the abovewaveform averaging has been done to constrain dependence to 3 successivechips. It may appear that the 4-chip dependence at point X of FIG. 13has not been entirely eliminated. However, the correlation function ofthis waveform is shown in FIG. 14, and indicates that indeed there is aprecipitous drop on the chip dependence beyond +/−1 chip. Substantially,3 chip-only dependence is maintained for target echoes with up to +/−0.5chip mis-sampling.

The apparent failure to get rid of the anomalies at location X in FIG. 6is an artifact of the graph plotting program. There are in fact only 2possible waveform points on each side on each anomaly, but the graphplotting program draws lines joining either one of the leftmost pointsto either one of the rightmost points by linear extrapolation.Nevertheless, a receive filter would do precisely that, and so it isdesirable to remove this discontinuity. This arises because the threechips on which the leftmost waveform values depend are for exampleb2,b3,b4 and then this changes suddenly to a dependence on b3,b4 and b5,as they are now the nearest chips to the rightmost point. In order toavoid this discontinuity, those specific points at location X of theanomaly may depend only on the overlapping symbols b3 and b4 and may notdepend on either the oldest bit b2 of the previous symbol shift nor thenewest bit b5 of the subsequent symbol shift, but must be converged to atotal of four points depending only on b3 and b4.

The effect of such a waveform discontinuity is clearly seen in thespectrum of FIG. 15. While the correlation function has been improved,the far-out spectral sidelobes have risen from a −150 dB level to a −70dB level. This is to some extent inevitable, as the power spectrum isthe Fourier transform of the autocorrelation function; however,eliminating the discontinuity should not degrade the correlationfunction but should improve the spectrum.

While producing the waveform of FIG. 13 was readily automated,eliminating the discontinuity in FIG. 13 is truly more akin tohandcrafting. The reason is, if the points at location X of the anomalyare replaced with their average, this will cause a discontinuity withthe preceding and following points. Therefore, a smooth modification ofthe curves along their whole length is required to force convergence atthe four points X. This may be achieved by the following procedure.

In step 1, two values at the left-hand side of the anomaly aredesignated as a1 and b1 and those on the right are designated as a2 andb2. The values of notional points midway between the left and rightpoints on their respective waveforms are computed as a_(1.5)=(a1+a2)/2;and b_(1.5)=(b1+b2)/2.

In step 2, if the value of a_(1.5) is the greater and the value ofb_(1.5) is the smaller, the factor 1+α is computed, by which waveform amust be reduced and waveform b increased at that notional point to forceconvergence, as:

(1+α)b _(1.5) =a _(1.5)/(1+α)   (4)

α=√(a _(1.5) /b _(1.5))−1

Now, it is desired that the above factor should modify the waveform atthe desired point of convergence, but that the factor should graduallydiminish to unity at the center of the eye and at the ends where thewaveforms are already acceptable.

This is done by applying a factor 1+0.5α(1−cos(θ)) to modify thewaveform points, where θ varies from 0 to 180 degrees along the waveformfrom each end to the middle. This factor is unity at the ends and in themiddle, but is the desired factor 1+α at the anomaly.

FIG. 16 illustrates an exemplary correlation function of the waveformwhen handcrafted as explained above. The chip-dependence outside of +/−1bit descends even more precipitously than before. FIG. 17 confirms thatthe discontinuity has been removed by handcrafting, and FIG. 18 showsthat the far-out spectrum has been reduced about 10 dB. Of course, thefar-out spectrum may be further reduced by the low-pass roofing filters(1060A,B) with the reintroduction of some sidelobes of the correlationfunction. To avoid such sidelobes becoming troublesome, the filtercutoff frequencies should be, for example, Gaussian filters with a BT inthe 1.5 to 3 range. A sharp filter will produce more ringing andsimulation has shown that a sharp cutoff of about 3 times the bitratekeeps the ringing on the ACF down to about the −70 dB level.

If it is desired to further reduce the correlation sidelobes, one methodof achieving this is to increase the BT of the GSMK modulation. However,this is a straight choice between spectral sidelobes and correlationsidelobes. The resulting waveforms must be handcrafted anew for eachchoice, and sufficient information has been disclosed above for a personskilled in the art to analyze such a choice for a particularapplication. Attention is thus now turned to alternative waveforms thatcan be useful in an exemplary automotive MIMO radar system, and whichmay reduce correlation sidelobes further while achieving a bettercompromise with spectral sidelobes than GMSK.

GMSK waveforms have a 3-symbol dependence because of the 8 trajectorywaveforms that may be seen in FIGS. 6, 12, and 17, even at the optimumsampling point. GSMK has that characteristic because an I-bit lyingbetween two Q bits of equal polarity cannot achieve full amplitude asthere is no Q zero crossing between the two equal Q-bits. If Q isnon-zero, I cannot be unity due the constant envelope constraint whereI²+Q²=1. The peak value in the center of the I-bit may in fact bepredicted to be:

Ipeak=+1, if(Q1.xor.Q2)=1   (5a)

Ipeak=+√(1−|Qmin|²), if(Q1.xor.Q2)=0   (5b)

In order to have single-chip dependence in the middle of the eye, Qminmust therefore be zero, that is, the Q waveform should go to zerobetween two Q bits, even when they are the same.

FIG. 19 introduces some new diagrams that assist in understandingconstant-envelope, digital modulation.

The trellis diagram in FIG. 19 indicates how the phase changes when thefrequency is modulated according to the frequency modulation waveformillustrated below the trellis diagram. Assuming that the phase at pointQ−1 is −90 degrees, the first frequency step up to +dF causes the phaseto change at the rate of 2π.dF radians per second. If dF=0.25 B=0.25/T,where B is the bit rate or chip rate and T is the reciprocal of B,namely the chip period, then the phase change over one chip period fromQ−1 to Io is exactly 90 degrees, so that the phase moves from −90 to 0from point Q−1 to Io. If Q1 is also encoded into a +dF frequency shift,the frequency remains at dF for a further time T and the phase changesby a further 90 degrees to the value +90 degrees at point Q1. Theconstellation diagram shows by asterisks where the phase ends up atpoints I0,Q1,I2,Q3, etc. Points denoted by In have terminal phases thatare either 0 or 180, where the real part of the complex vector is +/−1corresponding to the maximum eye-opening of the real part I, while theimaginary part goes to zero corresponding to a zero crossing of theimaginary part, Q. Correspondingly, constellation points denoted by Qnhave terminal phases that are either +90 or −90, and the I-values go tozero at those points.

In FIG. 19, the frequency deviation is either +dF or −dF and changesabruptly from one value to the other, while the phase changes at aconstant rate from one constellation point to the next. This modulationis known as “minimum shift keying” (MSK), and has the eye diagramillustrated in FIG. 20 and has the spectrum of FIG. 21. The spectralsidelobes are seen to be of the order of 15-20 dB higher than those ofthe handcrafted GMSK modulation illustrated in FIG. 18, due to theabsence of filtering to round the waveform transitions. The eye diagramillustrates a small anomaly at zero crossings which is partly anartifact of the graph-plotting program interpolating between the finitenumber of samples (16 in FIG. 20) used per bit period. Nevertheless,this is exactly what will occur in a receiver bandpass filter andaffects the resulting correlation function. The correlation function istherefore plotted in FIG. 22 with the number of samples per chip varyingfrom 8 to 256, in factors of 2. It can be seen that the limit, for alarge number of samples per bit, corresponding more closely to acontinuous waveform, is a correlation function that falls precipitouslyto zero at +/−1 chip offset. With precise mid-eye sampling, the waveformtherefore depends on only one chip, but with mis-sampling it depends ontwo adjacent chips, one I-chip and one Q-chip, as may be seen in the eyediagram of FIG. 20. This is a reduction of GMSK's 3-chip dependence withmid-eye sampling and 4-chip dependence (or 2-chip, with handcrafting)with mis-sampling, although this only seems to be achieved with a largenumber of samples per bit. Using 8 samples per chip, and with ½-chipmis-sampling, the waveform may be seen from the correlation functionvalue at +/−1.5 chips to depend on two additional bits that have aninfluence at the −60 dB level. This drops to −70 dB using 16 samples perchip and continues to reduce with greater numbers of samples per chip.However, further research shows that the skirts of the correlationfunction with small numbers of samples per bit were due to slightnumerical inaccuracies that made the waveforms slightly different independence on past history. Handcrafting by averaging the correspondingpoints in the eye diagram, storing the averaged values in a waveformlookup table (e.g. memory 1030 of FIG. 5), and then using the averagepoints, results in the sidelobe skirts being eradicated, which isillustrated in FIG. 27 for 8 samples per chip and in FIG. 28 for as fewas 4 samples/chip.

It is desirable to reduce the spectral sidelobes far away from the mainlobe to a level lower than what unfiltered MSK achieves. This should beaccomplished by not changing the frequency abruptly between +dF and −dFbut rather by using a smoother transition. If smoother transitions areproduced by low-pass filtering the frequency modulating waveform, thisis tantamount to using GMSK and will re-introduce additional intersymbolinterference (ISI or correlation sidelobes). To obtain a differentresult, shaping is used rather than filtering. The shaped waveform canbe made the same for each chip and independent of the value of apreceding or following chip, thereby achieving spectral improvementwithout the addition of correlation sidelobes (aka ISI). Moreover, toensure that the phase ends up at the same constellation points of +/−90or 0/180 after each chip, and not a value depending on chip history, thearea under the shaped frequency waveform must remain the same value ofdF×T=0.25. FIG. 23 illustrates that a raised cosine frequency pulse thatpeaks at 2 dF has this property. FIG. 24 illustrates the eye diagram forthis modulation and FIG. 25 illustrates the spectrum, demonstrating themore rapid fall of the spectral sidelobes as a result of the raisedcosine shaping. The correlation function before handcrafting thewaveform is illustrated for 16 samples per chip in FIG. 26. Afterhandcrafting the waveform by averaging all corresponding points to forma waveform memory and using the waveforms from memory, the correlationfunction remains substantially ideal down to as few as 4 samples perchip, as illustrated in FIG. 29.

The effect of receiver filtering when the transmitter uses handcraftedraised cosine digital FM is now illustrated in FIGS. 30 and 31 tocompare with using GMSK, which was illustrated in FIGS. 11 and 12. Thepractical significance of these parameters is as follows: If the radarsystem needs to implement strong target subtraction in order to unmaskweaker target reflections that are close in both range and Doppler, thenthe complexity of the strong target cancellation procedure isproportional to the number of correlation sidelobes after the receivefilter that are significantly strong. For example, if it is desired tocancel a strong target echo to a level of −60 dB relative its uncanceledvalue, then FIGS. 30 and 31 illustrate that, for a receiver filter BTfactor of 0.53 that gives a 1 dB loss of signal-to-noise ratio comparedto a matched filter, the correlation sidelobes at +/−1.5 chips are at alevel of approximately −62 dB. Thus, canceling the principal lobe andsidelobes at +/−1 chip will result in a 62 dB suppression even with themaximum +/−0.5 chip mis-sampling. Compared with FIGS. 11 and 12, theamount of suppression using GMSK would be approximately 48 dB. Thus, theraised cosine modulation provides a significant improvement over GMSK asregards to strong target suppression with a 3-tap interference canceler.

Other receiver filters can be considered, such as Boxcar filters, Besselfilters and the like, however, the present invention is more concernedwith determining an optimum transmitter modulation. The transmittermodulation performances have therefore been compared using the samerange of receiver filter characteristics, typified by a Gaussian filterwith a range of −3 dB points relative to the chiprate determined byusing various BT factors.

The choice of a raised cosine, as mentioned above, was arbitrarily basedon it being a known smooth function. A consideration may also be made asto what other properties such a function should have for a radarapplication, with a view to producing optimized properties. FIG. 32illustrates the shape of the exemplary class of functions to beinvestigated. The function is zero for a bit interval of 0.5<x<−0.5,will have unity area, so as to equate to a unit area impulse in terms ofthe phase change it will cause when used as a frequency modulatingwaveform, and will have as many zero derivatives as possible at +/−0.5.As the function will be multiplied by the bit polarities, when the bitpolarity changes at x=+0.5, any non-zero derivative at that point flipsin sign, exhibiting a step, and the next highest order derivative willtherefore exhibit impulses. The ultimate rate of fall off of spectralsidelobes is 6N db/octave when N is the order of the derivative of thewaveform at which impulses first appear. Impulses have a flat spectrum,so working backwards, integrating N times will produce an increase inspectral roll-off slope by an extra 6 db/octave for each integration.Therefore it is desired that the function should have as many zeroderivatives as possible at x==+0.5 in order to avoid discontinuitieswhen the function is flipped by the random sign of a code chip.

Therefore, given the criteria a function f(x) should satisfy, namely:

-   -   a. (i) Unit area when integrated from x=−0.5 to x=+0.5.    -   b. (ii) As many zero derivatives as possible at x=+0.5, and        optionally    -   c. (iii) a prescribed not-to-be-exceeded value at x=0,        a set of simultaneous equations can be solved for the        coefficients of a polynomial in (x/2)² that meet the above        criteria, of which a few are plotted in FIG. 33.

The polynomials found by the above method were of the form

ao+a1x ² +a ² x ⁴ +a ³ x ⁶   (6)

with the following coefficients:

a0 a1 a2 a3 a4 zero order 1 first order 1.5 −1.5 second 1.875 −3.751.875 order third order 2.1875 −6.5625 6.5625 −2.1875 fourth 2.46094−9.4375 14.76563 −9.4375 2.46094 order

FIG. 33 illustrates the shaping functions of second, third and fourthorder that are computed without constraint (iii) above. However, ifconstraint (iii) is applied, curves with a peak frequency deviationlimited to different values can be obtained for each order by giving upone of the zero derivative constraints (ii).

The effect on eye pattern, correlation sidelobes, and spectral sidelobesmay also be explored to determine shaping functions that may be betterin a given application (such as exemplified by the raised cosine shapediscussed herein). However, the over-exploration of different functionshas limited merit because such a function will only be used for alimited number of samples per bit (e.g., 4, 8 or 16, and those 4, 8 or16 values are going to be quantized to a limited number of bits ofaccuracy). When using N samples per bit, the spectrum is only definedout to +/−N/2 chiprates, and thus, small differences in the functionsthat cause higher or lower far-out spectral sidelobes may be masked bythe time and value quantizations. Attention is thus turned to the effectof value quantization of the I,Q waveforms and methods to determine thebest quantized values and how to digital-to-analog convert them.

When I,Q values of digital FM signals are computed to a high degree ofaccuracy, a constant envelope will be maintained, namely I²+Q²=1. Whenhowever, the I and Q values are quantized to integer values less thansome maximum value, such as +/−31 for 6-bit quantizing, +/−63 for 7-bitquantizing, or +/−127 for 8-bit quantizing, it is not possible toguarantee that the squares of all pairs of integers sum to the sameinteger value, and thus, a constant envelope cannot be maintainedexactly, and the quantized I,Q values will have both amplitude and phaseerrors. However, since the transmit power amplifier is hard limiting,the amplitude errors will be substantially shaved off, leaving only thephase errors. Therefore, in one exemplary embodiment, a higher prioritywhen selecting quantized I,Q values is given to pairs of values that areclosest in phase to the unquantized vector without regard to amplitudeerror. Too much amplitude error may not be acceptable, but seeking pairsof quantized I,Q values (Kx,Ky) that are closest in phase angle, whereKx is within +/−1 quantizing step of the closest quantizing to I and Kyis within +/−1 quantizing step of the closest quantizing to Q, shouldgive lower spectral sidelobes after hard limiting. This was confirmed tobe so. However, there are low-pass filters (1060A,B) after thedigital-to-analog converters (1050A,B) of FIG. 5 that alter the modifiedI,Q values, so it is necessary to compute the merit of this alternativequantization after these filters are included. This was done usingGaussian filters with a −3 dB cutoff point of 2 bitrates, and it wasfound that it was still beneficial to search for quantized values within+/−1 LSB that best matched the phase of I,Q.

Mathematically, integer values (Kx′,Ky′) within +/−1 of (Kx,Ky) aresought for which:

(Kx′/Rk−I/R)²+(Ky′/Rk−Q/R)² is a minimum

where Rk=√(Kx′²+Kx′²) and R=√(I²+Q²).

The modified values (Kx′,Ky′) were found to give lower spectralsidelobes after filters (1060A,B) and hard limiting in the transmitpower amplifier.

Another practical imperfection that can give rise to elevated spectralsidelobes is digital-to-analog converter (DAC) accuracy. If the DAC doesnot give equal steps, this can result in additional quantizing noise. Inparticular, if a strong signal cancellation unit attempts to mimic thequantizing in the transmitter in order to maximize cancellation, thedifferences between the transmit DAC and the model used for cancellationwill result in less effective cancellation. To mitigate this, a specialform of DAC may be used that effectively guarantees equal quantizingsteps in the mean, and this is briefly described below.

An exemplary 8-bit DAC (1050A,B) comprises 256 nominally equal currentsources, each of which can be turned on and off by logic fed via an8-bit value. When the 8-bit value is zero, no current sources are turnedon, and when the 8-bit value is 255, 255 current sources are turned on(with one current source remaining turned off). In one exemplaryembodiment, the 256 current sources are arranged in a ring, with thosethat are turned on occupying a first segment of the circle and thosethat are turned off occupying the other part of the circle. Whenever anew 8-bit value is received, it is first determined whether more currentsources will be turned on, or whether current sources will be turnedoff. If more current sources are to be turned on, the additional one(s)of the currently OFF current sources clockwise of the ON segment areturned ON, while if fewer current sources are to be ON, then currentsources counterclockwise of the currently ON segment are turned OFF. Inthis way, the ON segment of current sources and the OFF segment ofcurrent sources continuously rotate giving all current sources equal usein the mean for contributing to every desired analog value. Moreover,the time between a current turning on and off is maximized, thusreducing the effect of any speed limitations. In this way, the errorspectrum in the mean is zero and is reduced for lower frequencies sothat the error power spectrum is quadratic rather than flat with reducedtotal net error power. In one exemplary embodiment, thedigital-to-analog converter (DAC) is also balanced, like much of therest of the high-frequency circuitry for the reasons mentioned above. Abalanced DAC would transfer a current from a “+output” to a “−output” independence on the digital value, thus providing a bipolar conversionwith the digital value represented in the difference of the currents atthe + and −outputs.

Other methods for optimizing the spectrum can be attempted when thenumber of samples per bit used to represent shaped digital FM signals issmall. For example, at 4 samples per bit, if the vector is starting outat the 90 degree position [(I,Q)=(0,1)] and heading for the 0 degreeposition [(I,Q)=(1,0)] in four steps, the first sample is (0.1), themiddle sample, no. 3, is at 45 degrees (0.7071,0.7071), and the 5thsample is (1,0). Only samples 2 and 4 remain to be defined and symmetrydictates that sample 2 is the same angular displacement from 90 degreesas sample 4 is from 0 degrees. Therefore, there is only one variablethat can be explored to reduce spectral sidelobes.

Alternatively, the sample instants can be displaced by half a sampleeither side of the +/−90 and 0/180 points. The displacement is then afirst variable and the angular position of the samples on either side ofthe 45 degree points is then a second variable. The spectrum can now beexplored and optimized as a function of those two variables. Thecorrelation function may be broader for this alternative, however, asthe constellation points all depend on two bits and never only one.

Yet again, it can be beneficial if the vector dwells for two samplesaround each of the +/−90 and 0/180 degree points. The latter case wasinvestigated and the values optimized for best spectrum, giving thefollowing I,Q values quantized to 8 bits accuracy are listed below:

I Q 0 127 52 117 117 52 127 0

The above values are used either with a + or − sign depending on the Iand Q bit polarities required. It will be seen that the values (0,127)or (127,0) are repeated twice at the junction of two successive bits.

When sampling as coarse as 4 samples per bit, it is necessary to have asharper cutoff in the post DAC filters to suppress sidelobes beyond +/−2bitrates; for example, a Gaussian −3 dB cutoff in the region of 0.8bitrates. Moreover, after hard limiting in the power amplifier, thecorrelation sidelobes were improved compared to using higher cutofffrequencies. The eye diagram, when using the above I,Q values, isillustrated in FIG. 35. The spectrum with a Gaussian post DAC filter BTfactor of 0.8 is illustrated in FIG. 36, while the correlation sidelobesare illustrated in FIG. 37.

With 8 samples per bit, if the starting and ending sample values are asabove at 90 degrees and 0 degrees respectively, then the middle sample 5will be 45 degrees and samples 2, 3, and 4 will be at the same anglesfrom 90 degrees, as samples 8, 6, and 7 will be from 0 degrees, so thereare three variables to explore. Defining the quantity to be optimized,for example, as the total spectral energy beyond an exemplary +/−1.5bitrates, it is within the computational capabilities of a PC to explorethis as a function of three variables within a reasonable time if sodesired.

Several methods have been discussed for optimizing a constant envelopemodulation for use in a millimeter wave digital FMCW automotive radarwith regard to the parameters that are important in such a system. Themodulation is defined by a limited number of I,Q samples per bit, suchas 4, 8, or 16, that are quantized in an optimum manner to a limitedword length of, for example, 6, 7, or 8 bits. The I,Q samples are storedin memory (1030) from where they are recalled in dependence on thepolarity of the modulation bits from a code generator, which may beprecoded, and with regard to the current angular quadrant. Precoding andkeeping track of the quadrant is performed by the state machine of I,Qselection logic (1020). The selected quantized I,Q samples aredigital-to-analog converted using the above described analog-to-digitalconversion techniques that shape the digital to analog quantizationerror noise to facilitate accurate subtraction of strong target echoesin the receiver by using a replica of the transmit modulator to generatea delayed, phase changed, and amplitude-weighted version that bestmatches the signal to be subtracted. The digital-to-analog convertedanalog signals are low-pass filtered by post digital-to-analog filters(1060A,B) and then a radar carrier signal is quadrature modulated at adesired center or mean frequency.

The exemplary embodiments disclosed herein cover many variations ofconstant envelope signals including GMSK, Raised Cosine shaped pulse-FM,and polynomial shaped pulse FM, and a person of normal skill in the artcan derive many other variations using the principles exposed hereinwithout departing from the spirit and scope of the invention asdescribed by the attached claims.

1. A method for determining a frequency modulation of a continuous waveradar for a vehicle, the method comprising: generating a symbol streamcomprising a time-sequence of modulating symbols, each symbol belongingto a limited set of symbols having unique numerical values; feeding thesymbol stream through one of (i) a shaping and (ii) filtering process toobtain a sequence of shaped or filtered symbols, respectively,comprising multiple sample values per symbol period, wherein the samplevalues represent samples of the shaped or filtered symbols at instantsseparated by intervals of a fraction of a time period between successivesymbols; calculating, for the sample instants, (I,Q) samples of I and Qwaveforms resulting from frequency modulating a carrier signal with thesequence of modulating symbols, wherein each I value is equal to acosine of resulting instantaneous carrier phases at the sample instants,and each Q value is equal to a sine of the instantaneous carrier phasesat the sample instants; for each possible set of values of a group ofsuccessive symbols on which a waveform is to depend, producing anaverage waveform over all symbol values outside the group; and on whichthe waveform is not to depend, superimposing all waveforms within plusand minus half a symbol period of the center symbol of each group havingthe same set of values and averaging the superimposed I, Q samples toproduce for each group an averaged set of I, Q samples, and an averagewaveform; and recording final I, Q values to be stored in a memory forsubsequent use in producing the frequency modulation in the frequencymodulated continuous wave radar.
 2. The method of claim 1 furthercomprising rotating the phase angle of all the I, Q samples by an amountthat brings an eye diagram into a focus that best represents themodulating symbols.
 3. The method of claim 1 further comprising, whenthe average waveform produced by a group of symbols exhibits adiscontinuity with the average waveform produced by a following set ofsymbols having the same overlapping symbols as a previous symbol shift,erasing the discontinuity by applying a smoothing function that forcesthe sample values on either side of the discontinuity to converge whilehaving an effect on sample values that diminishes towards zero furtherfrom the discontinuity.
 4. The method of claim 1, wherein thetime-sequence of modulating symbols is a sequence of binary bits.
 5. Themethod of claim 4, wherein the sequence of binary bits is a maximumlength sequence.
 6. The method of claim 1 further comprising precodingthe symbol stream according to a precoding scheme for an intendedreceiver to produce a precoded symbol stream, such that the symbolstream is a precoded symbol stream.
 7. The method of claim 6, whereinthe modulating symbols are binary bits, and wherein the precodingcomprising modulo-2 adding each bit with its predecessor to produce theprecoded symbol stream.
 8. The method of claim 1, wherein the feedingthe symbol stream through one of (i) a shaping and (ii) a filteringprocess to obtain a sequence of shaped or filtered symbols,respectively, comprises feeding the symbol stream through a Gaussianfilter.
 9. The method of claim 1, wherein said feeding the symbol streamthrough one of (i) a shaping and (ii) a filtering process to obtain asequence of shaped or filtered symbols, respectively, comprisesmultiplying each symbol value by a shaping function to obtain a pulseshaped symbol waveform.
 10. A radar system for a vehicle, the radarsystem comprising: at least one transmitter configured for installationand use on a vehicle, and configured to transmit a frequency modulatedradio signal; at least one receiver configured for installation and useon the vehicle, and configured to receive radio signals that include thetransmitted radio signals transmitted by the transmitters and reflectedfrom objects in an environment; wherein a first transmitter of the atleast one transmitter comprises a modulator comprising a code generator,a waveform selection module, a filter module, and a memory; wherein thecode generator is configured to generate a symbol stream comprising atime-sequence of modulating symbols, each symbol belonging to a limitedset of symbols having unique numerical values; wherein the filter moduleis configured to filter the generated symbol stream to generate asequence of filtered symbols comprising multiple sample values persymbol period, wherein the sample values represent samples of thefiltered symbols at instants separated by intervals of a fraction of atime period between successive symbols; wherein the waveform selectionmodule is configured to calculate, for the sample instants, (I,Q)samples of I and Q waveforms resulting from frequency modulating acarrier signal with the sequence of modulating symbols, wherein each Ivalue is equal to a cosine of resulting instantaneous carrier phases atthe sample instants, and each Q value is equal to a sine of theinstantaneous carrier phases at the sample instants; wherein thewaveform selection module is configured to, for each possible set ofvalues of a group of successive symbols on which a waveform is todepend, produce an average waveform over all symbol values outside thegroup; and on which the waveform is not to depend, to superimpose allwaveforms within plus and minus half a symbol period of the centersymbol of each group having the same set of values and to average thesuperimposed I, Q samples to produce for each group an averaged set ofI, Q samples, and an average waveform; and wherein the waveformselection module is configured to record final I, Q values to the memoryfor subsequent use by the modulator in producing the frequency modulatedcontinuous wave signal.
 11. The radar system of claim 10, wherein thewaveform selection module is further configured to rotate the phaseangle of all the I,Q samples by an amount that brings an eye diagraminto a focus that best represents the modulating symbols.
 12. The radarsystem of claim 10, when the average waveform produced by a group ofsymbols exhibits a discontinuity with the average waveform produced by afollowing set of symbols having the same overlapping symbols as aprevious symbol shift, which erases the discontinuity via theapplication of a smoothing function that forces the sample values oneither side of the discontinuity to converge while having an effect onsample values that diminishes towards zero further from thediscontinuity.
 13. The radar system of claim 10, wherein thetime-sequence of modulating symbols is a sequence of binary bits. 14.The radar system of claim 13, wherein the sequence of binary bits is amaximum length sequence.
 15. The radar system of claim 10, wherein thecode generator is configured to generate the symbol stream according toa precoding scheme for an intended receiver to produce a precoded symbolstream, such that the symbol stream is a precoded symbol stream.
 16. Theradar system of claim 15, wherein the modulating symbols are binarybits, and wherein the precoding comprising modulo-2 adding each bit withits predecessor to produce the precoded symbol stream.
 17. The radarsystem of claim 10, wherein the filter module comprises a Gaussianfilter.
 18. The radar system of claim 10, wherein the filter module isconfigured to multiply each symbol value by a shaping function to obtaina pulsed shaped symbol waveform.